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On Hilbert’s irreducibility theorem

Volume 180 / 2017

Abel Castillo, Rainer Dietmann Acta Arithmetica 180 (2017), 1-14 MSC: 11C08, 11G35, 11R32, 11R45. DOI: 10.4064/aa8380-2-2017 Published online: 9 August 2017

Abstract

We obtain new quantitative forms of Hilbert’s irreducibility theorem. In particular, we show that if $f(X, T_1, \ldots, T_s)$ is an irreducible polynomial with integer coefficients, having Galois group $G$ over the function field $\mathbb Q(T_1, \ldots, T_s)$, and $K$ is any subgroup of $G$, then there are at most $O_{f, \varepsilon}(H^{s-1+|G/K|^{-1}+\varepsilon})$ specialisations $\mathbf{t} \in \mathbb Z^s$ with $|\mathbf{t}| \le H$ such that the resulting polynomial $f(X)$ has Galois group $K$ over the rationals.

Authors

  • Abel CastilloDepartment of Mathematics, Statistics,
    and Computer Science
    University of Illinois at Chicago
    851 S Morgan St
    Chicago, IL 60607, U.S.A.
    e-mail
  • Rainer DietmannDepartment of Mathematics
    Royal Holloway
    University of London
    TW20 0EX Egham, United Kingdom
    e-mail

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